On classification of the solutions for general elliptic equation

Abstract: The B\"acklund transformations and the superposition formulas for two sub--equations of the general elliptic equation are constructed from the Riccati equation by using an indirect mapping method.The thirty-six previously known solutions of the general elliptic equation are proved to be equivalent to the another ten solutions.The classification of solutions for the general elliptic equation is obtained based on the equivalence relations.The general elliptic equation expansion method with new classified solutions are used to obtain abundant new exact traveling wave solutions of a modified Camassa-Holm equation.